Let R be a two-dimensional normal graded ring over a field of characteristic p>0. We want to describe the tight closure of (O) in the local cohomology module HR+2(R) using the graded module structure of HR+2(R). For this purpose we explore the condition that the Frobenius map F: [HR+2 (R)]n→[HR+2 (R)]pninduced on graded pieces of HR+2(R) is injective. This problem is treated geometrically as follows: There exists an ample fractional divisor D on X=Proj (R) such that R=R (X, D)= ⊕n≥0H0(XOX (n D)). Then the above map is identified with the induced Frobenius on the cohomology groups[Figure not available: see fulltext.] Our interest is the case n<0, and in this case, a generalization of Tango's method for integral divisors enables us to show that Fn is injective if p is greater than a certain bound given explicitly by X and nD. This result is useful to study F-rationality of R. The notion of F-rational rings in characteristic p>0 is defined via tight closure and is expected to characterize rational singularities. We ask if a modulo p reduction of a rational signularity in characteristic 0 is F-rational for p≫0. Our result answers to this question affirmatively and also sheds light to behavior of F-rationality in small p.
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